Compare the graphs of the logarithimic functions f(x)=log7x and g(x)=log4x. For what values of x is f=g, f>g, and f<g? explain how you know
The graphs of logarithmic functions have some distinct characteristics.
For f(x) = log7x, the base of the logarithm is 7, which means we are taking the logarithm of x in base 7. This function will have a vertical asymptote at x = 0 and will increase as x increases. It will intersect the y-axis at (1, 0).
Similarly, for g(x) = log4x, the base of the logarithm is 4. This function will also have a vertical asymptote at x = 0 but will increase at a slower rate than f(x). It will intersect the y-axis at (1, 0).
To determine when f(x) = g(x), we need to set the two functions equal to each other: log7x = log4x. To simplify this equation, we can use the property of logarithms that says if log_a(b) = log_a(c), then b = c. Applying this property, we can write 7x = 4x.
Solving this equation, we get x = 0. This means that f(x) and g(x) intersect at x = 0.
To determine when f(x) > g(x), we need to compare the values of f(x) and g(x) for different values of x. Since f(x) increases at a faster rate than g(x), we can conclude that f(x) > g(x) for all x > 0.
Similarly, f(x) < g(x) for all x < 0, as g(x) increases more slowly than f(x).
In summary:
- f(x) = g(x) at x = 0.
- f(x) > g(x) for all x > 0.
- f(x) < g(x) for all x < 0.
These conclusions are based on the properties of logarithmic functions and their respective bases.